Note that, since \(M\in \operatorname{mod}_0(R)\) , we may assume \(d\ge 2\) . Note also that, the case where \(n=0\)  is [1]}. Moreover, the case where \(n=1\)  is a consequence of Corollary REF (ii). Hence we may assume \(n\ge 2\) .
If \(n\)  is odd, then the assertion follows from Corollary REF . Next assume \(n\)  is even. Then Proposition REF  implies that \(\operatorname{H}^{d-n+1}_{\mathfrak {m}}(M\otimes _RM^*)=0\) . As \(M\)  is reflexive, \(M\approx M^{**}\approx \Omega ^2\operatorname{Tr}M^*\) . Therefore, by Lemma REF , we have that \(\operatorname{Ext}^{d-n}_R(M,M)\cong \operatorname{Ext}^{d-n+2}_R(\operatorname{Tr}M^*,M)=0\) . Consequently, as \(d-n\)  is even, we conclude that \(\operatorname{pd}_R(M)<\infty \) , i.e., \(M\)  is free; see [2]}.
